calculate the 16th term of an arithmatic progression he its fifth term is 6 and its 12th is 41.

a+4d = 6

a+11d = 41

7d = 35
d = 5
a = -14

-14 + 15(5) = 61

To find the 16th term of an arithmetic progression, we can use the formula:

an = a1 + (n - 1)d

where:
an is the nth term,
a1 is the first term, and
d is the common difference.

First, we need to find the value of a1 (first term) and d (common difference).

Given that the fifth term (a5) is 6 and the twelfth term (a12) is 41, we can set up the following equations:

a5 = a1 + 4d
a12 = a1 + 11d

Since the fifth term is given as 6, we can substitute this into the first equation:

6 = a1 + 4d Equation 1

Similarly, we can substitute the twelfth term of 41 into the second equation:

41 = a1 + 11d Equation 2

Now we have a system of two equations with two variables. We can solve this system to find the values of a1 and d.

First, let's solve Equation 1 for a1:

a1 = 6 - 4d

Now substitute this value of a1 into Equation 2:

41 = (6 - 4d) + 11d

Simplifying the equation:

41 = 6 + 7d

Subtract 6 from both sides:

35 = 7d

Divide both sides by 7:

d = 5

Now we have the value of d, which is the common difference. We can substitute this back into Equation 1 to find a1:

6 = a1 + 4(5)
6 = a1 + 20

Subtract 20 from both sides:

a1 = -14

Now that we have the values of a1 and d, we can use the formula to find the 16th term (a16):

a16 = a1 + (16 - 1)d
a16 = -14 + 15(5)
a16 = -14 + 75
a16 = 61

Therefore, the 16th term of the arithmetic progression is 61.